Bellman filtering for state-space models

This article presents a filter for state-space models based on Bellman's dynamic programming principle applied to the mode estimator. The proposed Bellman filter (BF) generalises the Kalman filter (KF) including its extended and iterated versions, while remaining equally inexpensive computationally. The BF is also (unlike the KF) robust under heavy-tailed observation noise and applicable to a wider range of (nonlinear and non-Gaussian) models, involving e.g. count, intensity, duration, volatility and dependence. (Hyper)parameters are estimated by numerically maximising a BF-implied log-likelihood decomposition, which is an alternative to the classic prediction-error decomposition for linear Gaussian models. Simulation studies reveal that the BF performs on par with (or even outperforms) state-of-the-art importance-sampling techniques, while requiring a fraction of the computational cost, being straightforward to implement and offering full scalability to higher dimensional state spaces.Comment: 24 page

Brownian motion and multidimensional decision making

This thesis consists of three self-contained parts, each with its own abstract, body, references and page numbering. Part I, "Potential theory, path integrals and the Laplacian of the indicator", finds the transition density of absorbed or reflected Brownian motion in a d-dimensional domain as a Feynman-Kac functional involving the Laplacian of the indicator, thereby relating the hitherto unrelated fields of classical potential theory and path integrals. Part II, "The problem of alternatives", considers parallel investment in alternative technologies or drugs developed over time, where there can be only one winner. Parallel investment accelerates the search for the winner, and increases the winner's expected performance, but is also costly. To determine which candidates show sufficient performance and/or promise, we find an integral equation for the boundary of the optimal continuation region. Part III, "Optimal support for renewable deployment", considers the role of government subsidies for renewable technologies. Rapidly diminishing subsidies are cheaper for taxpayers, but could prematurely kill otherwise successful technologies. By contrast, high subsidies are not only expensive but can also prop up uneconomical technologies. To analyse this trade-off we present a new model for technology learning that makes capacity expansion endogenous. There are two reasons for this standalone structure. First, the target readership is divergent. Part I concerns mathematical physics, Part II operations research, and Part III policy. Readers interested in specific parts can thus read these in isolation. Those interested in the thesis as a whole may prefer to read the three introductions first. Second, the separate parts are only partially interconnected. Each uses some theory from the preceding part, but not all of it; e.g. Part II uses only a subset of the theory from Part I. The quickest route to Part III is therefore not through the entirety of the preceding parts. Furthermore, those instances where results from previous parts are used are clearly indicated.Research support from the Electricity Policy Research Group in Cambridge is gratefully acknowledged (http://www.eprg.group.cam.ac.uk)

Irreversible investment under predictable growth: Why land stays vacant when housing demand is booming

The standard model of irreversible investment under uncertainty considers only the level of the cash flow that could be obtained through the investment. We present a general model that includes as state variables both the level and the growth rate of the cash flow, while the timing and size of the one-time investment are discretionary. As an illustration, we consider an investor with the exclusive right to develop a vacant piece of land, where the timing of the investment and the scale of the property are chosen optimally. We demonstrate that construction is optimally postponed when prospects are gloomy, but also when they are bright. Indeed, under sufficiently high growth it is, perversely, never optimal to invest. Under a cost-of-capital argument, the rational response to predictable growth combined with flexible investment conditions is to keep land vacant for extended periods, which may explain why construction in superstar cities often appears sluggish. Our proposed model can be used in all investment decisions, irrespective of sector, where the assumptions of predictable growth and a one-off, flexible but otherwise irreversible investment are met

Distribution theory for Schr\"odinger's integral equation

Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schr\"odinger's equation. This paper, in contrast, investigates the integral form of Schr\"odinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schr\"odinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schr\"odinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's result to hypersurfaces. Second, we derive a new closed-form solution to Schr\"odinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schr\"odinger's differential equation. Third, we derive boundary conditions for `super-singular' potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution, and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schr\"odinger's integral equation is viable tool for studying singular interactions in quantum mechanics.Comment: 23 page

When Is Information Sufficient for Action? Search with Unreliable yet Informative Intelligence

We analyze a variant of the whereabouts search problem, in which a searcher looks for a target hiding in one of n possible locations. Unlike in the classic version, our searcher does not pursue the target by actively moving from one location to the next. Instead, the searcher receives a stream of intelligence about the location of the target. At any time, the searcher can engage the location he thinks contains the target or wait for more intelligence. The searcher incurs costs when he engages the wrong location, based on insufficient intelligence, or waits too long in the hopes of gaining better situational awareness, which allows the target to either execute his plot or disappear. We formulate the searcher’s decision as an optimal stopping problem and establish conditions for optimally executing this search-and-interdict mission

Can Google search Data help predict macroeconomic series?

We make use of Google search data in an attempt to predict unemployment, CPI and consumer confidence for the US, UK, Canada, Germany and Japan. Google search queries have previously proven valuable in predicting macroeconomic variables in an in-sample context. However, to the best of our knowledge, the more challenging question of whether such data have out-of-sample predictive value has not yet been answered satisfactorily. We focus on out-of-sample nowcasting, and extend the Bayesian structural time series model using the Hamiltonian sampler for variable selection. We find that the search data retain their value in an out-of-sample predictive context for unemployment, but not for CPI or consumer confidence. It is possible that online search behaviours are a relatively reliable gauge of an individual’s personal situation (employment status), but less reliable when it comes to variables that are unknown to the individual (CPI) or too general to be linked to specific search terms (consumer confidence)

When is Information Sufficient for Action? Search with Unreliable Yet Informative Intelligence

The article of record may be found at: http://dx.doi.org/10.1287/opre.2016.1488We analyze a variant of the whereabouts search problem, in which a searcher looks for a target hiding in one of n possible locations. Unlike in the classic version, our searcher does not pursue the target by actively moving from one location to the next. Instead, the searcher receives a stream of intelligence about the location of the target. At any time, the searcher can engage the location he thinks contains the target or wait for more intelligence. The searcher incurs costs when he engages the wrong location, based on insufficient intelligence, or waits too long in the hopes of gaining better situational awareness, which allows the target to either execute his plot or disappear. We formulate the searcher’s decision as an optimal stopping problem and establish conditions for optimally executing this search-and-interdict mission

Bellman filtering and smoothing for state-space models

This paper presents a new filter for state–space models based on Bellman’s dynamic-programming principle, allowing for nonlinearity, non-Gaussianity and degeneracy in the observation and/or state-transition equations. The resulting Bellman filter is a direct generalisation of the (iterated and extended) Kalman filter, enabling scalability to higher dimensions while remaining computationally inexpensive. It can also be extended to enable smoothing. Under suitable conditions, the Bellman-filtered states are stable over time and contractive towards a region around the true state at every time step. Static (hyper)parameters are estimated by maximising a filter-implied pseudo log-likelihood decomposition. In univariate simulation studies, the Bellman filter performs on par with state-of-the-art simulation-based techniques at a fraction of the computational cost. In two empirical applications, involving up to 150 spatial dimensions or highly degenerate/nonlinear state dynamics, the Bellman filter outperforms competing methods in both accuracy and speed